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Exact Trigonometry Table for all Angles

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Exact Trigonometry Table for all Angles

[Precise –Rewritten]

Bhava Nath Dahal

This book on series with:

Exact Values in Trigonometry: Five New Techniques

Publisher: CreateSpace Independent Publishing Platform

ISBN –10: 1536995002

ISBN –13: 978 –1536995008

Kindle ASIN B01KFP3560

Series – 3

Copyright © 2016 Bhava Nath Dahal

All rights reserved.

Introduction

We can compute exact trigonometric values using Precise –Rewritten method. The detail of the Precise –Rewritten method, its basis, proof of method and examples of exact values of all integer angles and polygon has described either in Exact Values in Trigonometry: Five New Methods (Vol 1) or in Precise –Rewritten Method: Exact Values in Trigonometry (Vol 2). In this brief works, list the exact values of trigonometry mainly Sine of all integer angles and polygon up to 24 –gon has described.

Precise –Rewritten method is based on chord (notation as ‘a’) and supplementary chord (notation as ‘b’). We compute chord (a) of double angle of given angle. From that chord (a), compute supplementary chord (b) using:

b = √(4 – a2)

Because of easy pattern of Precise-Rewritten method, ‘b’ will have just sign changes after the first radical from negative ‘-ve’ to positive ‘+ve’.

Trigonometric Formulae

Based on academic, technical and practical use, there are six main trigonometric ratios. There are variety of ways to establish the relations. As we already discussed, Precise –Rewritten method of exact value of trigonometry is based on chord (a), supplementary chord (b) and their product (ab).

Main relations:

#

Sin A =ab/2

#

Csc A = 2/ab

#

Cos A = 1 – a2/2

#

Sec A = 2/(2 –a2)

#

Tan A = ab/(2 –a2)

#

Cot A = (2 –a2)/ab

Verse and Coverse relations:

Practically in less use, there are further six trigonometric ratios. They are two in each sector of Verse –based, Coverse –based and Ex –based. Their chord –based formulae are as follows:

#

Versin A= a2/2

#

Vercos A= b2/2

#

Coversin A = 1 – ab/2

#

Covercos A=1 – ab/2

#

Exsec A = a2/( 2 –a2)

#

Excsc A = (2 –ab)/ab

Haverse and Hacoverse relations:

Practically almost no use, there are further four trigonometric ratios. They are half of Verse –based and Coverse –based formula. Their chord –based formulae are as follows:

#

Haversine = a2/4

#

Havercosine = b2/4

#

Hacoversine = (2 –ab)/4

#

Hacovercosine = (2+ab)/4

If we know the length of chord, half of that chord (a/2) will be the Sine of half of angle representing that chord (not included above). This is the magical development of determination of Sine of an angle since last 2000 years.

Exact Trigonometric Values

Following is the list of Sine of integer angles (A). All of the result of Sin A has divided by 2. Double of Sin A is chord (a) of double angle (2A).

For example, Sin 1˚ is ½ √(2 – √(2 + √(2 [+ + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ++ √(2 – + √(2 +]]

[Please care the sign of repetition has underlined. Closing brackets are collapsed for easy.]

Double angle for 1˚ is 2˚ and chord (a) for 2˚ is √(2 – √(2 + √(2 [+ + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ++ √(2 – + √(2 +]].

What’s about supplementary chord (b) for 2˚?

This is simple, just change the first sign (which is –ve in all case) into +ve. Therefore, supplementary chord for 2˚ is √(2 + √(2 + √(2 [+ + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ++ √(2 – + √(2 +]].

From these ‘a’ and ‘b’, we can compute any trigonometric ratios as given in the formulae list above.

What’s about the complementary angle? The answer is simple, it is based on supplementary chord (b). Therefore, Sin 89˚ is ½ √(2 + √(2 + √(2 [+ + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ++ √(2 – + √(2 +]].

Sin 1˚= ½ √(2 – √(2 + √(2

[+ + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ++ √(2 – + √(2 +]]

[Please care the sign of repetition has underlined.]

All integer angles Sin 1˚ – Sin 45˚

[Please care the sign of repetition has underlined. Closing brackets have collapsed by ‘]’ for easy.]

A˚ Sin A˚

1˚ ½ √(2 – √(2 + √(2 [++ √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2+] ]

2˚ ½ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

3˚ ½ √(2 – √(2 + √(2 [++ √(2 + √(2 – √(2 – √(2+] ]

4˚ √(2 – √(2 [++ √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2+]]

5˚ ½ √(2 – √(2 + √(2 [++ √(2 + √(2 – √(2+]]

6˚ ½ √(2 – √(2 + √(2 + √(2 – √(2]

7˚ ½ √(2 – √(2 + √(2 [++ √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2+]]

8˚ ½ √(2 – √(2 [++ √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2+] ]

9 ½ √(2 – √(2 + √(2 [++ √(2 – √(2+] ]

10 ½ [++ √(2 – √(2 + √(2+] ]

11 √(2 – √(2 + √(2 [++ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2+] ]

12 ½ √(2 – √(2 [++ √(2 – √(2 – √(2 + √(2 ]+]

13 ½ √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 ]

14 ½ √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

15 ½ √(2 – √(2 + √(2 – √(2]

16 ½ √(2 – √(2 [++ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2+] ]

17 ½ √(2 – √(2 + √(2 [+ – √(2+ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 +] ]

18 ½ [++ √(2 – √(2+] ]

19 ½ √(2 – √(2 + √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

20 ½ √(2 – √(2 [++ √(2 – √(2 + √(2+] ]

21 ½ √(2 – √(2 + √(2 – √(2 + √(2 + √(2 – √(2 – √(2 + √(2 + √(2 – √(2 ]

Easy alternative is ½ √(2 – √(2 + √(2 – √(2 [++ √(2 + √(2 – √(2 – √(2+] ]

22 ½ √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 ]

23 ½ √(2 – √(2 – √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

24 ½ √(2 – √(2 – √(2 – √(2 + √(2 + √(2 ]

25 ½ √(2 – √(2 – √(2 – √(2 + √(2 + √(2 ]

26 ½ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

27 ½ √(2 – √(2 – √(2 – √(2 + √(2 ]

28 ½ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 ]

29 ½ √(2 – √(2 – √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

30 ½ √(2 – √(2]

31 ½ √(2 – √(2 – √(2 – √(2 – √(2 [++ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2+] ]

32 ½ √(2 – √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

33 ½ √(2 – √(2 – √(2 – √(2 – √(2 + √(2 + √(2 – √(2 – √(2 + √(2 + √(2]

34 ½ √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 ]

35 ½ √(2 – √(2 – √(2 [++ √(2 – √(2 + √(2 + √(2 – √(2 + √(2+] ]

36 ½ √(2 – √(2 – √(2 [++ √(2 – √(2+] ]

37 ½ √(2 – √(2 – √(2 [++ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2+] ]

38 ½ √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 ]

39 ½ √(2 – √(2 – √(2 [++ √(2 – √(2 – √(2 + √(2+] ]

40 ½ √(2 – √(2 – √(2 + √(2 + √(2 ]

41 ½ √(2 – √(2 – √(2 [+ + √(2+ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 +] ]

42 ½ √(2 – √(2 – √(2 + √(2 + √(2– √(2 ]

43 ½ √(2 – √(2 – √(2 [++ √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2+] ]

44 ½ √(2 – √(2 – √(2 [++ √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2+] ]

45˚ ½ √2

46 ½ √(2 + √(2 – √(2 [++ √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2+] ]

47 ½ √(2 + √(2 – √(2 [++ √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2+] ]

48 ½ √(2 + √(2 – √(2 + √(2 + √(2– √(2 ]

49 ½ √(2 + √(2 – √(2 [+ + √(2+ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 +] ]

50 ½ √(2 + √(2 – √(2 + √(2 + √(2 ]

51 ½ √(2 + √(2 – √(2 [++ √(2 – √(2 – √(2 + √(2+] ]

52 ½ √(2 + √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 ]

53 ½ √(2 + √(2 – √(2 [++ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2+] ]

54 ½ √(2 + √(2 – √(2 [++ √(2 – √(2+] ]

55 ½ √(2 + √(2 – √(2 [++ √(2 – √(2 + √(2 + √(2 – √(2 + √(2+] ]

56 ½ √(2 + √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 ]

57 ½ √(2 + √(2 – √(2 – √(2 – √(2 + √(2 + √(2 – √(2 – √(2 + √(2 + √(2]

58 ½ √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

59 ½ √(2 + √(2 – √(2 – √(2 – √(2 [++ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2+] ]

60 ½ √(2 + √(2 – √(2]

61 ½ √(2 – √(2 – √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

62 ½ √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 ]

63 ½ √(2 + √(2 – √(2 – √(2 + √(2 ]

64 ½ √(2 + √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

65 ½ √(2 + √(2 – √(2 – √(2 + √(2 + √(2 ]

66 ½ √(2 + √(2 – √(2 – √(2 + √(2 + √(2 ]

67 ½ √(2 + √(2 – √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

68 ½ √(2 + √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 ]

69 ½ √(2 + √(2 + √(2 – √(2 + √(2 + √(2 – √(2 – √(2 + √(2 + √(2 – √(2 ]

Easy alternative is ½ √(2 + √(2 + √(2 – √(2 [++ √(2 + √(2 – √(2 – √(2+] ]

70 ½ √(2 + √(2 [++ √(2 – √(2 + √(2+] ]

71 ½ √(2 + √(2 + √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

72 ½ [++ √(2 – √(2+] ]

73 ½ √(2 + √(2 + √(2 [+ – √(2+ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 +] ]

74 ½ √(2 + √(2 [++ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2+] ]

75 ½ √(2 + √(2 + √(2 – √(2]

76 ½ √(2 + √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

77 ½ √(2 + √(2 + √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 ]

78 ½ √(2 + √(2 [++ √(2 – √(2 – √(2 + √(2 ]+]

79 √(2 + √(2 + √(2 [++ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2+] ]

80 ½ + √(2 + √(2 [+ + √(2+ √(2 – √(2 +] ]

81 ½ √(2 + √(2 + √(2 [++ √(2 – √(2+] ]

82˚ ½ √(2 + √(2 [++ √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2+] ]

83˚ ½ √(2 + √(2 + √(2 [++ √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2+]]

84˚ ½ √(2 – √(2 + √(2 + √(2 – √(2]

85˚ ½ √(2 + √(2 + √(2 [++ √(2 + √(2 – √(2+]]

86˚ √(2 + √(2 [++ √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2+]]

87˚ ½ √(2 + √(2 + √(2 [++ √(2 + √(2 – √(2 – √(2+] ]

88˚ ½ √(2 – – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

89˚ ½ √(2 + √(2 + √(2 [++ √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2+] ]

90˚ ½ 2 = 1

[Please care the sign of repetition has underlined. Closing brackets have collapsed by ‘]’ for easy.]

Exact Values of polygon chords

In the above sub –chapter we observed exact value of integer angles. In this sub –chapter, we shall discuss the exact length of few of the polygons using Precise –Rewritten method. Guru Euclid is father of polygonal geometry. No one Himafter developed such huge contribution on polygon yet. This sub –chapter is in His respect.

For any regular polygon, there is pattern for chord or length of a side. If quotient on dividing 360 by the given angle is an integer, that is regular polygon for this purpose. For example

360˚/1 ˚= 360 (this is 360 –gon, 1˚ has exact value in pattern)

360˚/18.94737˚= 19 (this is 19 –gon, 18.94737˚ has exact value in pattern)

=360˚/0.125874125874126˚= 2860 (this is 2860 –gon, 0.125874125874126˚ has exact value in pattern)

As a result, angle which is 360˚/n [where n is a natural] has exact value of trigonometry. Following is the length of a side of regular polygon up to n= 24.

N –gon Each side exact length

3 √(2 + √(2 ]

4 √2

5 √(2 – √(2 ]

6 √(2 ]

7 √(2 – ]

8 √(2 – √2]

9 √(2 – √(2 + √(2 – √(2 ]

10 √(2 ]

11 √(2 – √(2 ]

12 √(2 – √(2 + √(2 ]

13 √(2 – √(2 ]

14 √(2 – √(2 ]

15 √(2 – √(2 ]

16 √(2 – √(2 + √(2]

17 √(2 – √(2 + √(2 + √(2 ]

18 √(2 ]

19 √(2 – √(2 + √(2 ]

20 √(2 – √(2 + √(2 ]

21 √(2 – √(2 + √(2 ]

22 √(2 – √(2 ]

23 √(2 – √(2

24 √(2 – √(2 + √(2 + √(2 ]

[Closing brackets have collapsed for easy.]

How does above value determine?

A new method, named as Precise-Rewritten method for trigonometric values has used to determine above values. For detail of the methods, one may refer any of the following books as reference book:

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Exact Values in Trigonometry: Five New methods, printed by CreateSpace Independent Publishing Platform or a Kindle version on the same written by Bhava Nath Dahal, 2016.

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Precise-Rewritten Method: Exact Values in Trigonometry, printed by CreateSpace Independent Publishing Platform or a Kindle version on the same written by Bhava Nath Dahal, 2016.

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Método Precisa-Rewritten: Os valores exatos dentro Trigonometria, Kindle version, written by Bhava Nath Dahal, 2016.

Discuss regarding the method with the author

Scholar may approach to the author through email to [email protected] or through Facebook page as Bhava Nath Dahal or through YouTube channel as “Bhava Nath Dahal”.

One remarkable point, I am not a mathematician rather just an amateur mathematician. Please bring the discussion points strictly relating to new method for trigonometric values only.


Exact Trigonometry Table for all Angles

  • ISBN: 9781370077311
  • Author: Bhava Nath Dahal
  • Published: 2016-09-21 11:50:14
  • Words: 3264
Exact Trigonometry Table for all Angles Exact Trigonometry Table for all Angles